User blog:Holomanga/The Axiom of Union
We've explicitly constructed a set that has zero elements (the empty set), and thanks to the Axiom of Pairing we also have sets with one element and sets with two elements. But what about more? Certainly, set theory wouldn't be much good if it couldn't talk about sets of more than two things! Fortunately, there's an axiom for that, the Axiom of Union, one that allows you to construct new sets from existing sets. The Axiom of Union says that if you have a family of sets (a family of sets is just a name for a set that contains sets; "family" is used because set of sets sounds like you're some kind of Solomonic poet trying to apply emphasis), then you can construct another set - the union set - that contains elements that are in any of the sets in the family. The Axiom of Union is \forall X \exists A \forall x \left( x \in A \leftrightarrow \exists B \left( x \in B \wedge B \in X \right) \right) . A visual metaphor: a family of sets is a box filled with buckets, which each contain stuff. The union set is the bucket gained from pouring the contents of all of those buckets into one large bucket. For example, say you have four sets A , B , C , and D , and you want to make sure that the set \{ A, B, C, D \} exists. One could use the Axiom of Pairing to obtain the pairs \{ A, B\} and \{ C, D\} . Then one could apply the Axiom of Pairing again to get \left\{ \left\{ A, B \right\}, \left\{ C , D\right\} \right\} , and then the Axiom of Union on this family of sets to get the four-element set \left\{ A , B , C , D \right\} . The union of a family of sets is used so often that it deserves a special symbol. The union of the family of sets X is written as \cup X , and the union of a pair of sets A and B is written as A \cup B (read as "A union B"). This is the set of elements that are in A or B . Another set that we might want to construct, quite related, is the intersection. The union contains all of the elements that are in any of the sets in the family. The intersect contains all those elements that are in all of the sets in the family. Definining the intersect doesn't use its own axiom, because it's just a subset of the union so we can use the Axiom of Specification to pull it out. Specifically, \cap X = \left\{ x \in \cup X : \forall A \in X \left( x \in A \right) \right\} . When applied to a pair of sets, you can also write A \cap B . This is the set of elements that are in A and B . A third operation one might want to apply on a set is the difference of two sets, or the relative complement of one set in another. A \setminus B = \left\{ x \in A : x \notin B \right\} . This set, called the relative complement of B in A, is the set of all elements that are in A, but not in B. And, to complete the logical connectives, the symmetric difference between A and B is A\,\triangle\,B and defined as A \setminus B \cup B \setminus A , or equivalently A \cup B \setminus A \cap B . This gives the set of elements that are in A or B, but not both. These two definitions both describe the same set, as will be proven below. Since this is really neat, I'll do a table. Let P be the statement x \in A , and Q be the statement x \in B . Then we have the following logical statement sets: More ought to be said on the nature of the intersection of a family of sets. What we really want from an intersection of a family is analogous to what we want from the union of a family - the set of 'all' elements that satisfy \forall A \in X \left( x \in A \right) . I've pulled a bit of sleight-of-hand by saying that this is a subset of \cup X , and it might be worthwhile to try to back up that assertion. Let's suppose there's some larger set, Y , for which \cup X \subset Y , and see what happens when we pull the elements that satisfy the intersection criterion from it. For the below theorems, let Y be an arbitary proper superset of \cup X , and let I = \left\{ x \in Y : \forall A \in X \left( x \in A \right) \right\} The above proof seemed to rely on the assumption that X was not empty. What happens if we relax that assumption? Now, this is interesting! The "true intersect" of each set seems to be a subset of its union, except for the empty set, where it's potentially larger. How much larger? Well, it seems that because of the vacuous truth, 'any' element satisfies \forall A \in X \left( x \in A\right) , so by the nonexistence of the universal set there's no way to talk about the "true intersect" of the empty set! The solution to this is to swallow hard, ignore is, and say "suppose X is nonempty" at the start of all of our theorems about intersects. We can still say \cap \varnothing = \varnothing , of course (that's just the definition), but because of the above theorem it behaves slightly differentially to all other intersects so it's usually best to just ignore it. Category:Blog posts